The present invention concerns multiplex, or Fourier-transform, spectroscopy.
Fourier-transform spectroscopy is a way of adapting to light a Fourier-analysis technique commonly employed to determine the spectral content of an electrical signal. In accordance with the technique employed for electrical signals, the instantaneous value of the signal is repeatedly sampled at a rate at least twice the signal bandwidth, and the spectral content of the sampled signal is determined by performing a discrete Fourier transformation: ##EQU1## where u(k) is the value of the kth sample of the electrical signal and F(n) is the (complex) value of the nth spectral component.
A straightforward adaptation of this approach to light is not practical, because the electric field associated with visible light, for instance, is on the order of 10.sup.15 Hz, which dictates too short a sampling window for conventional sampling techniques. Moreover, even if short enough sampling windows were possible, it would be quite difficult to achieve the sampling rates that would be required for any meaningful bandwidth.
Fourier-transform spectrometers, also called multiplex spectrometers, overcome this difficulty by taking advantage of a special case of Parseval's theorem: ##EQU2## That is, if successive values of the autocorrelation of the incoming light can be determined, the power spectrum can be computed by taking the Fourier transform of the autocorrelation function.
FIG. 1 depicts a conventional Fourier-transform spectrometer, which makes such a determination. Incoming light, depicted in FIG. 1 as being produced by a point source 10 and transformed into a plane wave by a collimating lens 12, is divided by a beam splitter 14 between two paths. In the first path, to which the beam splitter 14 directs light by reflection, a first mirror 16 reflects light through the beam splitter to a detector 18. A focusing lens 20 typically concentrates the light onto the detector. In the second path, to which the beam splitter directs light by transmission, a second mirror 22 reflects the light to the beam splitter 14, which reflects it through the focusing lens 20 to the detector 18. In other words, the common detector 18 receives light from both paths.
We will assume that the difference between the optical lengths of the two paths is .DELTA.d. Accordingly, if Au(t) is the scalar representation of the field disturbance contributed by one path, the total field disturbance at the detector can be given by: ##EQU3## where c is the speed of light in a vacuum. The difference .DELTA.d is thus the difference between optical distances defined by the integrals of refractive index over the respective paths. This arrangement can produce an autocorrelation value because photodetectors respond to intensity rather than instantaneous field strength. That is, the photodetector 18 responds to the mean value of the square of the field-strength value given by equation (3), i.e., to the mean value of the following expression: ##EQU4## The mean values of the first two terms in expression (4) are the intensities that would result from the respective paths individually, while the mean value of the third term will be recognized as being proportional to the autocorrelation, evaluated at .DELTA.d/c, of the two field values.
A Fourier-transform spectrometer varies the distance .DELTA.d between the optical path lengths. Scanning mechanisms for optical distance variations can take many forms. One type, for instance, varies the pressure of a gas in the optical paths and thus varies the refractive index. Other types translate refractive wedges or pivot refractive slabs in one of the paths. FIG. 1 depicts yet another type, which comprises a drive mechanism 23 for moving one of the mirrors 16. The result of the path-difference variation is that the detector output as a function of time represents the sum of the individual intensities and the autocorrelation evaluated throughout a range of delays, and the Fourier-transform spectrometer obtains the spectrum by determining the Fourier transform of the (necessarily truncated) autocorrelation functions.
FIG. 1 depicts a typical device for performing the transformation digitally, although analog devices, such as compressive receivers, have also been proposed for performing the transformation. At equally spaced values of optical-distance difference, an analog-to-digital converter 24 is triggered to sample the output of the photodetector 18 and generate digital signals indicative of the sampled value. A typical way of achieving this equal-spaced triggering is to cause the output of an auxiliary HeNe or other stable gas laser (not shown) to propagate through the two interferometer paths to, say, a silicon photodiode whose output is fed to a Schmitt trigger, which triggers the analog-to-digital converter 24. Computation circuitry 26 performs a discrete Fourier transformation of the resulting "interferogram," and it employs an appropriate display mechanism 28 to present the thereby-determined spectrum to the user.
Since the intensities for the individual paths are not affected by the change in optical distance, the contribution to the photodetector output from the first two terms in equation (4) are the same in all of the samples taken by the analog-to-digital converter 24. It is only the third terms that is dependent on optical-path distance. As a result, the first term affects only the "zero-frequency" output of the Fourier-transformation process, so all of the remaining output terms represent the Fourier transform of the autocorrelation of the field-strength signal. As equation (2) indicates, these values are thus the squares of the absolute values of the corresponding terms in the Fourier transform of the field signal itself; that is, they represent the power spectrum of the received light. The spectral resolution is inversely proportional to the length of the scan, as it is in any correlation spectral determination method.
A review of the foregoing explanation reveals that the Fourier-transform spectrometer operates by converting the problem of sampling a signal at an impossibly high temporal frequency into that of sampling it at corresponding achievable spatial intervals. The required spatial frequency of sampling is determined by the spatial bandwidth of the incoming light in accordance with Shannon's sampling theorem: the sampling must occur at a spatial frequency at least twice the spatial bandwidth of the received light. For instance, if the received light is restricted to wavenumbers between 1000 cm.sup.-1 and 5000 cm.sup.-1, the sampling frequency must exceed one sample/micron (10,000 samples/cm) in order to avoid aliasing. The Fourier-transform spectrometer is practical because, unlike sampling at the corresponding temporal frequency, sampling at this spatial frequency is readily performed by employing mechanisms such as the auxiliary HeNe laser to provide the required sampling accuracy.
Unfortunately, although the desired positioning accuracy can readily be achieved through a limited range with linear drives, the scan time and physical volume of the interferometer tend to increase with scan length, and relatively large scan distances are necessary if fine wavelength resolution is to be obtained. For instance, in order to achieve a wavelength resolution of 1 cm.sup.-1 "unapodized," i.e., without a window function, the mirror needs to move 0.5 cm. And the required travel increases with the stringency of the resolution requirement. The expense of the drive increases disproportionately with the distance requirement, and for fine-resolution devices it can become a significant factor in the overall cost of the spectrometer. It would therefore be desirable to obtain greater resolution without the need for excessive mirror travel. This would also increase the rate at which spectra can be obtained in high-signal-to-noise cases.